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DP-GP-LVM A Bayesian Non-Parametric Model for Learning Multivariate Dependency Structures Andrew R. Lawrence 1 Carl Henrik Ek 2 Neill D. F. Campbell 1 1 University of Bath, UK 2 University of Bristol, UK Proceedings of the 36 th International

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DP-GP-LVM

A Bayesian Non-Parametric Model for Learning Multivariate Dependency Structures Andrew R. Lawrence1 Carl Henrik Ek2 Neill D. F. Campbell1

1University of Bath, UK 2University of Bristol, UK

Proceedings of the 36th International Conference on Machine Learning

Long Beach, California, USA 12 June 2019 Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 1 / 11

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Motivation

Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 2 / 11

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Motivation

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Motivation

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Motivation

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Background

Factorized Gaussian Process Latent Variable Model (GP-LVM) [1, 2]

X(1,2) X(1) X(2) F(1) F(2) Y(1) Y(2)

Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 4 / 11

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Background

Factorized Gaussian Process Latent Variable Model (GP-LVM) [1, 2]

X(1,2) X(1) X(2) F(1) F(2) Y(1) Y(2)

X(1,2,3) F(1) X(2,3) F(3) F(2) X(1,2) X(1,3) X(3) Y(3) X(2) Y(2) X(1) Y(1)

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Background

Manifold Relevance Determination (MRD) [3]

X F(2) F(1) F(T ) Y(1) Y(2) Y(T )

... ...

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Background

Manifold Relevance Determination (MRD) [3]

X F(2) F(1) F(T ) Y(1) Y(2) Y(T )

... ...

xn fn,t θt yn,d zd

n ∈ [1, N] t ∈ [1, T ] d ∈ [1, D]

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Background

Fully Independent MRD (fi-MRD) [3]

X f2 f1 fD y1 y2 yD

... ...

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Background

Fully Independent MRD (fi-MRD) [3]

X f2 f1 fD y1 y2 yD

... ...

xn fn,d θd yn,d zd

n ∈ [1, N] d ∈ [1, D]

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DP-GP-LVM

X y3 y2 y1 yD F(2) F(1) F(T )

... ...

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DP-GP-LVM

X y3 y2 y1 yD F(2) F(1) F(T )

... ... xn fn,t θt yn,d zd

n ∈ [1, N] t ∈ [1, T ] d ∈ [1, D]

DP

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Experiments

PoseTrack [4] – Two People

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Experiments

PoseTrack [4] – Two People

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Experiments

PoseTrack [4] – Two People

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Experiments

PoseTrack [4] – Two People

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Experiments

PoseTrack [4] – Missing Data

˜ N Missing (%) 10 10 10 20 20 20 ˜ D Missing (%) 10 20 30 10 20 30 PoseTrack 2 Person BGP-LVM [5] −50.02 ± 12.32 − 93.39 ± 18.84 −157.67 ± 32.20 −61.64 ± 13.95 −134.08 ± 22.73 −191.90 ± 33.57 fi-MRD [3] −40.62 ± 10.64 −109.73 ± 17.36 −138.67 ± 14.84 −54.92 ± 8.51 −149.36 ± 11.36 −248.89 ± 36.79 DP-GP-LVM −18.11 ± 0.48 − 35.83 ± 0.44 − 54.07 ± 0.49 −52.28 ± 0.29 −104.76 ± 0.47 −158.4 ± 0.81 PoseTrack 4 Person BGP-LVM [5] −121.06 ± 24.10 −189.52 ± 34.73 −358.36 ± 52.06 −102.98 ± 24.06 −209.79 ± 37.35 −322.79 ± 59.27 fi-MRD [3] −28.55 ± 0.73 −56.45 ± 1.34 −123.47 ± 27.63 −73.89 ± 1.23 −147.42 ± 2.03 −270.04 ± 45.07 DP-GP-LVM −28.14 ± 0.92 −55.44 ± 1.69 −107.41 ± 2.61 −71.80 ± 1.87 −143.06 ± 3.26 −311.39 ± 1.31 Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 9 / 11

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References

Carl Henrik Ek, J Rihan, Phil H. S. Torr, G Rogez, and Neil D Lawrence. Ambiguity modeling in latent spaces.

Int. Conference on Machine Learning for Multimodal Interaction, 2008.

Mathieu Salzmann, Carl Henrik Ek, Raquel Urtasun, and Trevor Darrell. Factorized orthogonal latent spaces. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 701–708, 2010. Andreas C. Damianou, Carl Henrik Ek, Michalis K. Titsias, and Neil D. Lawrence. Manifold relevance determination. In International Conference on Machine Learning (ICML), 2012.

M. Andriluka, U. Iqbal, E. Ensafutdinov, L. Pishchulin, A. Milan, J. Gall, and Schiele B.

PoseTrack: A benchmark for human pose estimation and tracking. In CVPR, 2018. Andreas C. Damianou, Michalis K. Titsias, and Neil D. Lawrence. Variational inference for latent variables and uncertain inputs in Gaussian processes. Journal of Machine Learning Research, 17(1):1425–1486, January 2016. Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 10 / 11

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Thank You

Poster #221

Pacific Ballroom - 12 June 2019 - 18:30–21:00

Acknowledgements

We would like to acknowledge the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 665992, the UK’s EPSRC Centre for Doctoral Training in Digital Entertainment (CDE, EP/L016540/1), the RCUK-funded Centre for the Analysis of Motion, Entertainment Research and Applications (CAMERA, EP/M023281/1), and the Royal Society for supporting this research. Lawrence, Ek, Campbell DP-GP-LVM ICML 2019 11 / 11